Yeah, that's about it, as I found too. Here is where I got on the proof.

(not far)

N must be even, else only odd divisors squared (odd) would add to even. So 1st 2 divisors are 1 and 2. The last two, a and b have a^2 + b^2 = N-5, which is odd. So only one of the squares is odd, and this means only one of a and b are odd. Calling d the even one then d might = 2e with e prime and after trying 2(3) we see 2(5) works. But why must e be prime is where I am stuck, Cheers